Topics in nonlinear and robust estimation theory

We propose new methods to improve nonlinear filtering and robust estimation algorithms. In the first part of the dissertation, we propose an approach to approximating the Chapman-Kolmogorov equation (CKE) for particle-based nonlinear filtering algorithms, using a new proposal distribution and the improved Fast Gauss Transform (IFGT). The new proposal distribution, used to obtain a Monte Carlo (MC) approximation of the CKE, is based on the proposal distribution found in the auxiliary marginal particle filter (AMPF). By using MC integration to approximate the integrals of the AMPF proposal distribution as well as the CKE, we demonstrate significant improvement in terms of both error and computation time. We consider the additive state noise case where the evaluation of the CKE is equivalent to performing kernel density estimation (KDE), thus fast methods such as the IFGT can be used. We also provide much improved performance bounds for the IFGT, and which unlike the previous bound, are consistent with the expectation that the error decreases as the truncation order of the IFGT increases. The experimental results show that we can obtain similar error to the sequential importance sampling (SIS) particle filter, while using fewer particles.;In the second part, we consider the problem of estimating a Gaussian random parameter vector that is observed through a linear transformation with added white Gaussian noise when there are both eigenvalue and elementwise uncertainties in the covariance matrix. When the covariance matrix is known then the solution to the problem is given by the minimum mean squared error (MMSE) estimator. Recently a minimax approach in which the estimator is chosen to minimize the worst case of two criteria called the difference regret [28] and the ratio regret [29] in an eigenvalue uncertainty region was proposed. A closed form solution was also presented under the assumption that the Gram matrix of the model matrix weighted by the inverse covariance matrix of the noise vector, and the random parameter's covariance matrix, are diagonalized by the same unitary matrix (we refer to this as the joint diagonalizability assumption). This assumption significantly limits the applicability of the estimator. In this work we present a new criterion for the minimax estimation problem which we call the generalized difference regret (GDR), and derive the minimax estimator which is based on the GDR criterion where the region of uncertainty is defined not only using upper and lower bounds on the eigenvalues of the parameter's covariance matrix, but also using upper and lower bounds on the individual elements of the covariance matrix itself. Furthermore the GDR estimator does not require the assumption of joint diagonalizability and it can be obtained efficiently using semidefinite programming. The experimental results show that we can obtain improved mean squared error (MSE) results compared to the MMSE estimator and the difference and ratio regret estimators.;Finally, we propose a new approach for robust parameter estimation under sensor positional uncertainty of parameters which are used as features for an unexploded ordnance (UXO) classification scheme. Obtaining better parameter estimates by addressing the uncertainty that may be present in the locations of the sensors, is important in order to obtain improved classification results. Future work will include simulation and validation of the new approach with a UXO classification scheme.